SU(2) is the 3-sphere: we can think of it as the unit sphere in
ℝ4, or ℂ2, or the quaternions ℍ.
The 120 points in the binary icosahedral group Γ ⊂ SU(2) are the vertices of a 4-dimensional regular polytope with 600 tetrahedral faces: the 600-cell.
You can learn more at the Wikipedia article on the 600-cell.
This animated gif of the 600-cell was created by Jason Hise and put into the public domain on Wikicommons.
